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Publication Detail
AGAINST CUMULATIVE TYPE THEORY
• Publication Type:
Journal article
• Authors:
BUTTON TIM, TRUEMAN R
• Publisher:
Cambridge University Press (CUP)
• Publication date:
12/2022
• Pagination:
907, 949
• Journal:
The Review of Symbolic Logic
• Volume:
15
• Issue:
4
• Status:
Published
• Print ISSN:
1755-0203
• Language:
en
Abstract
AbstractStandard Type Theory, ${\textrm {STT}}$ , tells us that $b^n(a^m)$ is well-formed iff $n=m+1$ . However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$ , which has more relaxed type-restrictions: according to $\textrm {CTT}$ , $b^\beta (a^\alpha )$ is well-formed iff $\beta>\alpha$ . In this paper, we set ourselves against $\textrm {CTT}$ . We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$ ’s type-restrictions are unjustifiable, the type-restrictions imposed by ${\textrm {STT}}$ are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for $\textrm {CTT}$ . We end by examining an alternative approach to cumulative types due to Florio and Jones [10]; we argue that their theory is best seen as a misleadingly formulated version of ${\textrm {STT}}$ .
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